In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short review of the most important results devoted to discrete counterparts of the maximum principle (called discrete maximum principles, DMPs), mainly in the framework of the finite element method, and also present our own recent results on DMPs for a class of second-order nonlinear elliptic problems with mixed boundary conditions.
One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.
The convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators, with focus on elliptic convection-diffusion operators in Sobolev space. Superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a Hilbert space. Then the same convergence result is verified for elliptic convection-diffusion equations using different preconditioning operators. The convergence factor involves the eigenvalues of the corresponding operators, for which an estimate is also given. The above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection-diffusion problems.
Discrete maximum principles are established for finite element approximations of nonlinear parabolic PDE systems with mixed boundary and interface conditions. The results are based on an algebraic discrete maximum principle for suitable ODE systems.
The subject of the paper is the mesh independent convergence of the preconditioned conjugate gradient method for nonsymmetric elliptic problems. The approach of equivalent operators is involved, in which one uses the discretization of another suitable elliptic operator as preconditioning matrix. By introducing the notion of compact-equivalent operators, it is proved that for a wide class of elliptic problems the superlinear convergence of the obtained PCGM is mesh independent under FEM discretizations, that is, the rate of superlinear convergence is given in the form of a sequence which is mesh independent and is determined only by the elliptic operators.
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